Geodesic
Integral Harnack inequality
Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 115-120

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Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc. 
Rao, Murali. Integral Harnack inequality. Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 115-120. doi: 10.1017/S0017089500005875
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